Basic Calculus refers to the simple applications of both differentiation and integration. Before delving into these processes, one must fully understand functions and their corresponding graphs. Y= f(x) would indicate that y is a function of x – this means that the quantity y is dependent on the quantity x, with the condition that one value of x corresponds to only one value of y. In this light, y=f(x) is the dependent variable, while x is the independent variable. Since every x has only one y, they can be considered as an ordered pair. The graph of the function is just an illustration of all the ordered pairs on a xy-plane.

The process of differentiation and integration deals with these functions to analyze the changing relationship between input and output. Consider the following function:

Y = 3x^2

The basic process of differentiation involves finding how y changes with respect to x. This rate of change is known as the derivative (dy/dx). To find the derivative, we differentiate both sides (on the right hand side multiply the coefficient by the exponent, and then subtract one from the exponent):

Y=3x^2

(d/dx)Y = (d/dx)3*x^2

dy/dx = 3*(d/dx)x^2

dy/dx = 3*(2x)

dy/dx = 6x

Thus, it can be seen that the derivative is 6x. So if the value of x is 1, then the rate of change is 6(1) = 6 units. On the other hand, the basic process of integration involves finding F from f(x). To find the integral, we integrate both sides (on the right hand side add one to the exponent, and then divide the function by the new exponent):

Y = 3x^2

F = int(3x^2)

F = (3x^3)/3

F = x^3

Since, integration deals with the accumulation of quantities, using the above function we can figure out the amount that has accumulated between two x values. For example between x=1 and x=3, the amount that has accumulated is 26 units for this given function:

F(3)-F(1) = 27-1 = 26

Thus, understanding basic calculus may prove to be a very practical tool to possess when dealing with changing scenarios.

Highland Appliance must determine how many color TV's and VCR's should be stocked. It cost highland $300 to purchase a color TV and $200 to purchase a VCR. A color TV requires 3 square yards of storage space. A VCR requires 1 square yard of storage space. The sale of a color TV earns Highland a profit of $150, and the sale o

Hi, please see the attached question which I am having trouble with and need help with.
I have done part a) and part b)
the E-L equation for part a) I got to be 2y''sinhx - 2y'coshx + ysinh^3(x) = 0
and for part b) the new variable I have is u = cosh x and with new limits u1 = cosh a and
u2 = cosh b
and for part c) I

FLATLAND the Movie, Area of a Fractal Plant
Arlene Square wants to add a special plant to her garden, but she is not sure if she has enough room for it to fit.
The plant starts as a square of side length 1 meter. Each year, the plant grows larger in a very unique way. The plant sprouts an additional square for every edge

Question #1:
The transformation formulas between Cartesian coordinates (x,y) and polar coordinates (r,θ) are as follows:
x = rcos(θ) y = rsin(θ)
r = sqrt(x2 + y2) tan(θ) = y/x, where you have to determine which quadrant θ is in.
Every θ has a reference angle α as defined on pp.721-722, 747 in the book.
r is a

The equation for radioactive decay is: N(t) = Noe-kt, where No is the initial number of radioactive atoms, N(t) is the number of radioactive atoms left after a time t, and k is the decay constant. The half-life of carbon 14 is about 5730 years.
Radioactive carbon dating makes the assumption that living things absorb carbon 14

A transcendental irrational number
S = sin0.2 rad
can be represented by the infinite series
S = sum (0 to infinity) of (-1)^n/[5^(2n+1) . (2n+1)!]
Let its partial sum be
SN = sum (0 to N) of (-1)^n/[5^(2n+1) . (2n+1)!]
a) Use the Ratio test to show the series converges absolutely.
b) Write down explicitly and compu

The common economic functions can be summarized as follows:
C(x) is the total cost of producing x items.
C(x)/x is the average cost per item if x items are made.
p(x) is the price per item when x items are sold.
R(x) = x*p(x) is the revenue collected when x items are sold.
P(x) = R(x)-C(x) is the profit made when x items

1. Create a function, f(x), and pick a point c such that the limit of f(x) as x approaches c from the right and the limit of f(x) as x approaches c from the left are equal and the function is continuous. Show the values of the limits and explain why the function is continuous. The explanation should be intuitive as well as mathe

The depth of water at the end of a pier varies with the tides. On a particular day, the low tides occur at 2:00 a.m. and 2:00 p.m. with a depth of 2.1 meters. The high tides occur at 8:00 a.m. and 8:00 p.m. with a depth of 6.3 meters. A large boat needs at least 4 meters of water to be safely secured at the end of the pier.
B

1. Consider the function below. By straightforward inspection (no differentiation is necessary), determine the coordinates (x,y) of the point of minimum. Further, by using the standard procedure involving differentiation, determine the coordinates (x,y) of all critical points (you are not required to classify which point is mini

1) Find the equilibrium solution of the differential equation:
Dy/dt = 3y(1-(1/2)y)
Sketch the slope field and use it to determine whether each equilibrium is stable or unstable.
2) Consider the initial value problem
Y^1 = 4 - y^2, y(0) = 1
Use the Euler's method with 5 steps to estimate y(1). Sketch the field and u

Question 1.
1) Find a vector normal to the surface z + 2xy = x2 + y2 at the point (1,1,0).
2) Determine if there are separable differential equations among the following ones and explain:
a) dy/dx=sin(xy),
b) dy/dx = (xy)/(X+y)
c) dr/d(theta) = (r^2+1)cos(theta)
3) Find the general solution of the differential

Question 1.
Find the critical points of the function
f(x, y) = 4xy2 - 4x2 - 2y2
and classify them as local maxima, minima or saddles or none of these.
Question 2.
The surface is defined by
z = 3x2 + 2y2 - 3
Find the equation of the tangent plane to the surface at the point (1, 1, 2).
Question 3.
(

1.Find the asymptotes, the intercepts and sketch the following function. A computer sketch is not sufficient. You must explain.For each of the following functions, do a complete analysis: determine the vertical and horizontal asymptotes, and the x and y intercepts; and sketch a graph of the function. Section 6.2, number 6
y

1. Apply the Basic Comparison Test (BCT) and Limit Comparison Test (LCT). Find an appropriate Comparison series to determine convergence or divergence. See attached file for formulas.
2. Apply the Integral Test: First use a method to see if the terms are decreasing, then to determine convergence or divergence. See attached f

18. Compute the volume of the solid formed by revolving the region bounded by about (a) the x-axis; (b) y = 4.
20. Compute the volume of the solid formed by revolving the region bounded by and about (a) the y-axis; (b) x = 1.
4. Sketch the region, draw in a typical shell, identify the radius and height of the shell, an

For a rectangular sperical triangle, where C = 90 degrees, show that:
tan a = sin b tan A
and,
tan b = sin a tan B
This is how far I have a gotten:
sin C = 1
cos C = 0
I am also aware that I should be using both the Law of Cosines and the Law of Sines. Using these, I was able to prove several other identities. I am co

Suppose total transportation cost for a product can be approximated by the function:
T(f) = 2.1f^2 - 25.9f +121.1
Where f is the number of facilities.
Ignoring any other costs, find the optimal number of facilities with the minimum total transportation cost.

Formulate but do not solve the following linear programming problem.
A travel company decides to advertise in the Saturday travel sections of two major newspapers in town. The advertisements are directed at three groups of potential customers. Each advertisement in newspaper A is seen by 60,000 group I customers, 35,000 group

Describe the following characteristics of the graph shown in Graph3.pdf:
1. Where is the function increasing, decreasing, or constant?
2. Are there any relative/absolute extrema? If so, where?
3. Is the graph smooth or choppy (piecewise)?
4. Are there any restrictions on the domain?
5. Are there any horizontal or

Based on the information, create a sketch of the function on the axes provided. Please provide detailed explanation. ** Please see the attached file for the complete problem description **
-Increasing and constant: none
-Decreasing: (-infinity, 2)... (please see the attached file)
Thanks!

SEE ATTACHMENT FOR ALL PROBLEM QUESTIONS
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